Calhoun: The NPS Institutional Archive

Theses and Dissertations Thesis Collection

1971-09

An algorithm for the solution of

concave-convex games

Gunn, Lee Fredric

Monterey, California ; Naval Postgraduate School

http://hdl.handle.net/10945/15562

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AN ALGORITHM FOR THE SOLUTIONOF CONCAVE -CONVEX GAMES

by

Lee Fredric Gunn

and

Peter Tocha

Thesis Advisor: J. M. Danskin, Jr.

September 1971

AppA.ovQ.ci faon. pubtic n.o.to.ai>2.; du>tsu,biutton Luitimltzd.

,0AM SCHOOL

An Algorithm for the Solution

of Concave- Convex Games

by

Lee Fredric GunnLieutenant, United States Navy

B.A., University of California, Los Angeles, 1965

and

Peter TochaKapitanleutnant , German Navy

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN OPERATIONS RESEARCH

from the

NAVAL POSTGRADUATE SCHOOL

I

ABSTRACT

A master's thesis which discusses the solution of con-

cave-convex games. An algorithm is developed, a computer

program written and applied to an anti-submarine warfare

force allocation problem as an illustration. Techniques

for handling concave -convex problems in high dimensions

are included.

TABLE OF CONTENTS

I. INTRODUCTION 5

A. A GAME THEORY APPROACH TO RESOURCEALLOCATIONS 5

B. CONTENT OF THE PAPER, BY CHAPTER 6

II. MATHEMATICAL CONCEPTS 7

A. GENERAL 7

B. THE ALGORITHM FOR THE SOLUTION OF CONCAVE-CONVEX GAMES 8

1. The Derivative Game 8

2. The Lemma of the Alternative 9

a. The Brown-Robinson Iterative Processin the "Auxiliary Game" 9

b. Statement of the Lemma 11

3. The Corollary of the Alternative 12

4. The Algorithm 13

III. PROGRAMMING ASPECTS ON AN ILLUSTRATIVE EXAMPLE - 18

A. FORMULATION OF THE PROBLEM 18

1. The Space X 20

2. The Space Y 20

3. The Function F(x,y) 21

B. BASIC COMPUTATIONS 2 2

1. Computation of 6, . 22r him

2. Partial Derivatives with Respect to x, .

-

23

3. The Value of F(x,y) 23

4. Partial Derivatives with Respect to Y-v" 23

5. Finding Directions of Increase(Decrease) 23

C. PROGRAMMING NOTES 24

1. Presentation of Input - The MissionDescription 24

2. The Contraction with Respect to V, . 27r him

3. Working in Subspaces 27

4. Machine Accuracy Problems 29

5. Testing the Program 30

6. General Comments 31

a. Initial Allocations 31

b. Distance Policy 32

c. Upper and Lower Bounds 32

d. Modification of the ObjectiveFunction 32

e. Assigning Values V, . 33& & himr r«1 • Z 1i i , i , 1 1 » w i > > ' ^ — —— — — — — — — — — — --~ — - - .... -\ •

IV. SUMMARY 35

V. SUGGESTIONS FOR FURTHER STUDY 36

APPENDIX: PROGRAMMING FLOWCHART 38

COMPUTER PROGRAM LISTING 51

BIBLIOGRAPHY 74

INITIAL DISTRIBUTION LIST 76

FORM DD 14 73 7 7

I. INTRODUCTION

A. A GAME THEORY APPROACH TO RESOURCE ALLOCATIONS

Problems in the allocation of resources can be divided

into two descriptive categories: Single-agent problems and

adversary problems. Single-agent problems have only one

participant optimizing without intelligent opposition. In

the solution of adversary problems, opponents work at cross

purposes. Each choice of an allocation of resources by one

participant must be made in light of those of his opponent (s).

Of course, games are adversary problems.

The great interest in game theory as a technique of

modelling which followed von Neumann's statement of the

fundamental concepts in 19Z7 [ISj ^oiilijiues today. The

fascination of the game as a model for conflicts of almost

any sort is enhanced by the fact that the solution to a

game is entirely independent of assumptions regarding the

actual behavior of the antagonist (s)

.

Matrix games and differential games are extensively

treated in the literature; some references are noted here.

Matrix games or games over the square are discussed in

basic form by Williams [21] and a thorough study is done by

Karlin [8]. For a first essay in the subject of differen-

tial games, see Isaacs [6]. Taylor [16,17] gives additional

examples of the modelling of combat operations including

search using differential games.

The development of the derivative game is due to

Danskin [4]. A very large class of concave-convex problems

yield to solution when the algorithm based on this game is

applied. It is with the derivative game and the algorithm

evolved from it that this paper is primarily concerned.

B. THE CONTENT OF THE PAPER, BY CHAPTER

Chapter II surveys some of the more important mathe-

matical ideas necessary to the development of the algo-

rithm for the solution of concave-convex games.

Chapter III is concerned with the programming of an

anti-submarine warfare force allocation example. The ex-

ample serves to illustrate both the use of the algorithm

and the theory on which it is founded. The problem is

formulated and the basic computational steps toward the

solution are enumerated. If any fears stem from the

formidable appearance of the matrices used in the example,

they hopefully will be dispelled by the description of the

form for input in the programming notes. Technical matters

regarding programming are considered in some detail.

Chapter IV draws together this presentation with some

concluding remarks

.

Chapter V contains suggestions for further study.

The Appendix consists of the programming flowchart.

The Computer Program Listing is included as well.

II. MATHEMATICAL CONCEPTS

A. GENERAL

The fundamental theorem of the theory of games in the

form appropriate here is the following:

Suppose F(x,y) is a continuous function on X x J,

where X and Y are compact and convex. Suppose that the set

of points X(y) yielding the maximum to F for fixed y is

convex for each such y, and that the set of points Y(x)

yielding the minimum to F for fixed x is convex for each

such x.

Then there exist pure strategy solutions x° and y°

satisfying

A 1 .'V - V I ' A I A * r! T7 v <, J.

(1)F(x,y°) < F(x°,y°), V xeZ.

A complete proof of the theorem in this form can be found

in [7]. Assuming that the conditions for the existence of

a solution satisfying (1) are met, the problem can be stated

in the form

Max Min F(x,y)

,

x y

which is equivalent to

Max cj)(x)

x

where <KX ) - Mi-n F( x >y)-

y

The theorem applies to two -person zero-sum games. Thus,

Max Min F(x,y) = Min Max F(x,y).x y y x

One important difficulty arises from the fact that,

although F may be smooth, cf> (x) is not in general differ-

entiate in the ordinary sense. Danskin, in [3], has shown

that under general conditions on X and Y, there exists a

directional derivative in every direction. It is this fact

that has provided the key to the solution of problems of

the type described.

B. THE ALGORITHM FOR THE SOLUTION OF CONCAVE -CONVEX GAMES

The algorithm to solve games concave in the maximizing

player and convex in the minimizing player is developed and

presented in great detail in [4]. The following gives a

brief survey of those results which are most important for

the design of the algorithm; to fill the apparent gaps, a

thorough reading of [4] remains necessary.

1 . The Derivative Game

Suppose x° cX . Associate with x° a non-empty set

of admissible directions y> r(x°) . W is defined as the

convex hull (e.g., see [19]) of the set of points

W(y) = {F (x°,y), -••, F (x°,y)},

xl yk

where yel(x°) .

The derivative game then is

H ( y , w ) = y * w

defined over r(x°) x W. The maximizing player maximizes H

by choice of yeT(x°), the minimizing player minimizes H by

choice of weW.

THEOREM I

A necessary and sufficient condition for the exis-

tence of a direction of increase for <J>(x) is that

the value of the derivative game defined by II be

positive at x° . The y° which yields the value of

the derivative game is a pure strategy.

If the value is positive one can find and use this direction.

If the value is non-positive, a direction of increase does

not exist, i.e., the solution has been reached.

The application of the derivative game in practice is greatly

complicated by the necessity for approximations.

2 . The Lemma of the Alternative

The Lemma takes into exact account the approxima-

te ons involved in the application of the derivative game.

It states that a certain process (to be explained below)

must either yield a sufficient increase to cj)(x) at a point

x° or determine that the point x° is nearly optimal. Be-

fore the lemma can be formulated in mathematical terms,

some further difficulties and the tools with which to over-

come them have to be outlined.

a. The Brown-Robinson Iterative Process in the"Auxiliary Game"

The Brown- Robinson (B-R) process employs the

following idea: Let G be the pay-off function. At stage

N=0 both players choose arbitrary strategies x° and y° . At

stage N=l the maximizer chooses x 1 such that G is maximized

against y° ; then the minimizer chooses y1 to minimize G

against x 1, and so forth. At stage N the maximizer chooses

x as if the minimizer ' s strategy were an evenly weighted

mixture of strategies y°,

••• • ,y]

; the minimizer chooses

y as if the maximizer' s strategy were an evenly weighted

mixture of strategies x° ,••• ,x .

IFor matrix games, Julia Robinson [12] proved

thatIN " 1 JN

lim supN-*-°°

r N-l N

m n=0 n=0

Danskin [1] has generalized the proof to hold

for two -person zero -sum games with continuous pay-off de-

fined over X x y, x and Y arbitrary compact spaces. It

should be noted here that the B-R process is very slow in

convergence when applied directly ^o finding an approxims

tion to the value of the game defined by (1). However, it

is not applied to the basic game in this algorithm but rather

to an "auxiliary game" for which an accurate solution is not

required.

The derivative game mentioned above cannot be

solved directly because the set Y(x°) is not known. All one

has is a single element ycY which approximately minimizes

F( x °>y)- The place of the derivative game, therefore, is

taken by the "auxiliary game" employing a modified version

of the B-R process described below. This process makes it

possible to keep track of the approximations involved and

their consequences. The "auxiliary game" is defined as

follows

:

10

For any £>0, denote by Y (x) the set of ycY

such that F(x,y) = <f>(x) + e- Let Y (5) = U Y (x) ,yeT^x ),

yeY(E).E(x°+d Y ,y) - <Kx°)

Then H( Y ,y) = ^ , for dQ

> 0,o

a minimum step size, is a game over r(x°) x Y (5) with y

the maximizing and y the minimizing player. This game has

optimal mixed strategies for both players. Applying the

idea of approximate optimization to the convergence proof in

[1] leads to

THEOREM II

N N-l nLet y be chosen such that ^ E n

H(y,y ) is

maximized to accuracy c, and y be chosen such

that rr Z^ H(y ,y) is minimized to accuracy n

.

i lien

lim sup \\ H( Y) yn

) - I En

H(Yn,y)

n=0 n=0< 2(c+n)

This holds for continuous H

Let 6 be the maximum oscillation of VF(x,y) over a distance

d . Theno

F(x°+do Y

N,y

n) - F(x°,yn ) > y

N F(x°,yn ) - e

_

o

The lemma of the alternative now can be formulated,

b. Statement of the Lemma

Suppose 0<a<3, y°eY (x°).

Then the generalized B-R process will, at some stage N, de

termine that one of the two following statements is true:

1. The maximum over F(x°) of the directional

derivative does not exceed 3.

11

N-N

2. The point x = x° + d y » where

N-N_l £ n

and the point y eY (x ) , where y minimizes

F(x ,y) to accuracy £, satisfy

F(xN,yN

) - F(x°,y°) 3

3> a bo ^- .

o o

3. The Corollary of the Alternative

Suppose that F(x,y) is concave in x and convex in

y. Then the modified B-R process applied at x° will, at

some stage N, determine that one of the two following state-

ments is true

:

1. The pair x = x°, y = y , where

-N _i_ <£ ny N+l L y

'

n=0

are approximate optimal strategies for the game

defined by F.

2. The point x zX yields an increase to <J>(x) by at

least a specified amount.

For details and proof see [4], pp. 36 ff.

Reference [4] continues with a detailed discussion

of delicate problems which can only be listed here: The

choice of the minimal step size d ; the problem of acces-

sibility of a point x from a point x° ; the problem of ob-

struction; the choice of a, 3, c, their interaction with each

other, and the choice of p where p is the accuracy to which

Max <t>(x) is to approximate the value of the game defined by

12

(1) . It must be noted that the conditions derived for the

selection of these parameters are sufficient.

4 . The Algorithm

The algorithm as a consequence of the foregoing

mathematical considerations is presented in section 10 and

11 of [4] and will not be reproduced here in detail. A ver-

bal description of its basic structure - depicted in Figure

1 -, however, may be useful:

The maximizing player, called Max, having arrived

at a point xx , has a direction of maximal increase y, ob-

tained either from the derivative game D cf>(xx) or from

the B-R process in the auxiliary game, and a distance d > d .

The minimizing player, called Min , is at a point yy. F(xx,yy)

is known. Max makes a proposal uo move to a point x = xx + dy

Of course, F(x,yy) > F(xx,yy). Min accepts Max's proposal

and starts minimizing against x, looking for a direction

of maximal decrease g. If there is none, yy is a minimum

against x as well as against xx in which case Max will move

to the point x. If there is a direction of decrease Min

forms a point y = yy + Dg such that F(x,y) < F(x,yy). A test

is performed to determine whether Min has already "beaten"

Max: if F(xx,yy) > F(x,y) Min stops the minimization pro-

cess, and Max discards his proposal x because moving to x

will not increase <K X ) • Max halves the distance d and, with

the same y, forms a new trial point x. If F(xx,yy) < F(x,y)

Min continues to minimize until either F(xx,yy) > F(x,y) or

Min can no longer find a direction of decrease. If now

13

Figure 1. Flowchart of Algorithm.

TRIAL X ANP YALLOCATIONS

YES

FAILURE.FOR X

->SUCCESS

FOR TRIAL X

TRIAL X bECO/AESMEW 5A5E POINT

FIND THE APPARENTDIRECTION OF FASTEST

INCREASE FOR X

YES

FORM A NEWTRIAL X

ADJUST XDISTANCE

YES

ENTER THE5-R PROCESS

PREFER TO FLOW CHARTOF b-R PROCESS)

THE PR06LEM|

i .» in i V i ii

CO TO STOP

|STOP

14

F(xx,yy) < F(x,y) , indicating that Min, even after a complete

minimization, was not able to "beat" Max, Max moves to the

proposed point x realizing a gain for <K X ) • Max then looks

for a new direction of increase. The process terminates when

such a direction does not exist.

The situation that leads into the Brown-Robinson

process is the following one: The proposed points x = xx+dy

have been "beaten" by Min until d gets cut down to d , in

spite of the fact that y, obtained from the derivative game,

is an apparently good direction. If the trial point x = xx

+ d y does not result in a move for Max, the B-R process,

Figure 2, is used.

Denote the present y - the one that so far has

lead to a failure rc-r r.:nx by y" , s^d liit; associated VF

' '' x

by to° . Minimize F completely against x = xx + d y . The

resulting y then leads to a new VF which is averaged with

the previous to,giving to

1. Suppose that to

1 as input to

the derivative game D 4>(x) produces a new y° such that the

value of the derivative game is positive as required. (If

such a y° does not exist the problem is solved). This y,

averaged with y1

,gives y

2 which in turn creates a new

trial point x 2 = xx + d y2

. x 2, or x

1

in general, then is

exposed to Min ' s reaction as described previously. Once Max

finds a direction and an associated trial point that cannot

be "beaten" by Min, Max moves and leaves the B-R routine.

The y-strategies are also averaged and saved although their

average is never used during the computation. In case the

15

Figure 2. The Brown-Robinson Process.

Conditions for entering the. E>-R

2At pt x° there was a direction t of

APPARENT INCREASE, HOWEVER, FAILURE AT

X1

- x°+ d y\ Have cj

START

NE>R-0

compute: y 1

= y 1

— O oCO =COY° =

NE>R=NbR+ 1 <

REFORM AVERAGE, Y, I e (THE SUPERSCRIPT N

CORRESPONDS TONE>R)

TRY XN= X + d T"

FINDMINIMIZING

CALCULATECO*

YES

REFORM OMEGA bAR

m (m-j)Cjw"'+lo

m

U)

GO TO D.F.

ALGORITHM

GET Yu + i

MOVE X

TO X*

D. F.

ALGORITHM

16

game defined by F(x,y) is terminated while in the B-R pro

cess, this average of the y-strategies represents the op-

timal solution for the minimizing player.

17

Ill . PROGRAMMING ASPECTS OF AN ILLUSTRATIVE EXAMPLE

A. FORMULATION OF THE PROBLEM

An algorithm for the solution of a wide class of con-

cave-convex games over polyhedra has been presented in

abbreviated form above. In this chapter that algorithm is

applied to a particular game, an anti-submarine warfare

force allocation problem.

For this problem, five indices are employed: h, i, j,

k, and m. In a meaningful example, the maximum numbers

corresponding to these indices might be, respectively:

5, 25, 10, 500, and 5. The indices have the following

meanings

:

): ; Submarine type

.1

k

Submarine mission

Type of antisubmarine weapon (or vehicle)

Place in which submarine and weapon encounter one

another

m: Stage of the submarine mission.

An additional index is used. &(k) is the "kind of place."

A "kind of place" might be defined by a particular set of

weapon employment parameters. These include the tactical

and the natural environment. The natural environment con-

sists of oceanographic and meteorological conditions. Ex-

amples of the tactical environment are destroyers in an ASW

screen and patrol aircraft in barrier patrol. The "kind of

place" in which an encounter occurs impacts on the outcome

18

of an encounter between submarine and weapon. A reasonable

number of "kinds of places" in the present context might be

25.

A submarine mission is described by a matrix |EU . ., I

.

' '

' hijkm '

'

This matrix has as its elements real numbers denoting the

extent to which a submarine of type h at stage m of mission

i is exposed to a weapon of type j at the kth place. The

effects of these weapons and thus of the encounters are

characterized by a "technical" matrix l^hi£fk")l ^n t ^ie ^°^"

lowing meaning: exp[-C, .

{,, >. ^'V^ ^- s tne probability that

a submarine of type h survives one exposure to y units of

weapons of type j at a "kind of place" £(k). These en-

counters are assumed to be mutually independent. Note that

the probability of survival to the m^n mission stage is

conditioned on the completion of the previous stages. Sup-

pose that there are y., units of force of type j at the ktn3 K

place. Then the probability of a submarine's completing

stage m of mission i is

7| exp[-Ehijkm , ChjA(k) yjk ],

m' <m

which, due to independence of the events, equals exp [-0, . ]

where

6him V *gEhijkm' Chj£(k) yjk*

m ' <m

Now, by carrying out a premultiplication

,

A = E Chijkm "hijkm hj£(k)'

19

the exponent becomes

6him= Z £

Ahijkm' y

jk'

in' <m

In this example, the vast majority of the E, . . , , and there

fore of the A, ..-, , are zero. If 3000 non-zero A,.., arehi j km hi j km

allowed, each type of submarine can be employed on ten dif-

ferent missions and undergo up to 60 encounters with anti-

submarine weapon systems.

1 . The Space X

Let x, . be the proportion of submarines of type h

assigned to the i"th mission. Make X =|

|x, .

|satisfy the

conditions

;OT pvprv h , x, ..hi ' ' hi -

1

and

ahi -xhi - 3hi'

£or ever >' P air h'i

'

where the sets {a, •}, (Bv-1 are supposed to satisfy

L, a,. < 1 < L, 3, • for every h

and

< a, - < 3^ • for every pair h,i.

2 . The Space Y

Let y . , be the proportion of antisubmarine forces

of type j sent to the k™ place. Make Y = l>'-vl satisfy

the conditions

20

and

L y-k

= 1, for every j, y^ k>

JK

ajk - yjk -

bjk

£or ever>r P air J> k

where the sets {a.-, }, {b., } are supposed to satisfy

L a., < 1 , < L b .,

k J k k J k

and

< a.-, < b . , for every pair j ,k

3 . The Function F(x,y)

V, • is the value of accomplishing stage m of

mission i for a submarine of type h. The character of

F(x,y) can be examined.

06 u

w, - = v, . exp r-e, .]him him * L him J

where 8, . is as before. Puthim

hi *-* himm

Thenm

F(x,y) = 7 xhi

Thi

.

h.,i

This function is linear in x and exponential in y and is

therefore a concave -convex game of the type treated in [4],

defined over X x Y. The quantity F(x,y) represents the

total expected payoff to the submarine player. Re-expressing

F(x,y) in its explicit form gives:

21

F(x,y) = L, l, x, . A, V, exp [- /. ), E, . .-, C, .. n sV., 1,v ,7 ^

i - hi ^ him v l v T hiikm h;i£(kyik J

111 m J k J j *, ; j

The remainder of this chapter gives details of the application

of the algorithm to the game defined above.

The principal result is the flow-chart (Appendix).

Since this flow chart is constructed around the algorithm

from [4], it is helpful, though not essential, to have [4]

available. The complete program listing is included following

the Appendix. The program is written in FORTRAN IV and was

run on the IBM 360/67 computer at the Naval Postgraduate

School, Monterey, California.

B. BASIC COMPUTATIONS

1 . C \_> nip u L a 1 1 o ii v f mm

For fixed (h,i), 9, . is non-decreasing in the mis-v'

J' him fa

sion stage m. This reflects the trivial fact that

P [submarine survives stage m]

< P [submarine survives stage m-1].

Equality holds when the "threat" due to the encounters at

stage m is non-existent, i.e., when either no ASW-forces

y., are present or their effectiveness against the submarine,

^h'&fkV ^" s zero - Hence 0, . , for each pair (h,i), is ac-

cumulated over the mission stages as follows:

6, . = G, .r ,. + V V A, . ., y ., , 9, . = .him hi (m-1) h* ^ hijkm 7 jk' hio

22

2. Partial Derivatives with Respect to x, .

hi

Because of the linearity of F in x the partials with

respect to x, . , F , are the coefficients of x, . and do notilJ- Xt z Jii

explicitly contain x. V}iim

exP[-6}lim ] can be represented as

a matrix of dimension (n x m) , where n is the number of pairs

(h,i). Then

Fv = V V, . exp[-0, .]x, . *-> him ^ L him J

is the sum of the elements in the (h,i) row of that matrix.

3. The Value of F(x,y)

F(x,y) is obtained by pre-multiplying F by x, . andXhi

ni

summing the products

F(x,y) - £ x,. Fv_ .

n i

4. Partial Derivatives with Respect to v.,* 7

-} K

Because of the cumulative property of 6, . , a change

in y-k

during some mission stage m ' will affect the following

stages as well. For each pair (h,i), premultiply x, . by

the corresponding Ahi -j km ,> where m' is the mission stage in

which the y., of interest occurs. Sum V, . expf-G, . 1 over1

K

him 'L him J

the mission stages for which m ' <m , multiply the result with

Xhi

Ahiikm'

and sum tne P roducts over h and i:

'\k~~ I ?Xhi

Ahi Jkm .

ra^ Vhim .

exp[-9him ,].

5• Finding Directions of Increase (Decrease )

F and F are inputs to the derivative games forhi } ik

x and y. For x, the direction y° is sought such that D <j>(x) =

VFx«y is maximized; for y, g° is sought such that D Max F(x,y)

g x

23

VF »g is minimized (equivalently , -VF «g is maximized). The

side condition is that y° and g° be unit vectors:

i k J 1 k J

A method of finding such directions - called THE DIRECTION

FINDING ALGORITHM - is derived from the Kuhn-Tucker conditions

and the Schwartz Inequality. It is contained in [4] and has

been applied to a vector - valued function by Zmuida in [22].

C. PROGRAMMING NOTES

The flow chart (Appendix) and the associated program may

not be optimal with respect to machine time and memory space

required, and may provide opportunities for improvement. For

computer routines like this, intended to solve high-dimensional

problems, a trade-off between time and space is generally ap-

parent. The user, considering the particular facilities

available to him, must decide on his optimal trade-off, and

modify the program accordingly. The following discusses some

of the techniques that have been implemented; it points out

the major difficulties that have been encountered and the ways

presently used to deal with these, and offers some remarks

about the impact of the underlying mathematics on the use of

the algorithm for realistic problems. The numbers referenced

are statement numbers.

1 . Presentation of Input - The Mission Description

In a realistic application, most of the A, • • •,, will

be zero because the effectiveness C, .

„

r , ^ contained in A willhj £(k)

be zero. Example: suppose k denotes a place in the western

Baltic and &(k) classifies this place as shallow with extreme-

ly poor sonar conditions; j denotes a nuclear killer-submarine,

1 A

h represents a conventional attack submarine. Then C, . . ,,.

will be zero for all practical purposes.

The method of presenting the matrix E which avoids

computing of and with A when C is zero, is one which is ex-

tremely easy for the user to understand and employ. He gives

his description of a submarine mission by plotting it on a

chart, marks off in order the places the submarine must go,

and notes the forces it might meet at those places. He

will give the exposure E required by the particular mission

in terms of a standard which he will have set. He will mark

off the points in the mission where the various stages of

the mission will have been accomplished. Such a description

might run as follows:

m=0

1% — 1 A — 1 1/-— "1 A — "B — ' Ma - _L 1-1 i\ ~ X J~ ^ J-. — X . U

j=5 E=1.5k=7 j = l E=2.0

j = 3 E=3.0

m=l (the first stage of the first mission is completed)

k=8 j=4 E=1.2j=7 E=1.7

m=2

k=15 j = 3 E=0.3j=2 E=2.0j=4 E=0.5

m=3

The above mission (mission 1 for submarines of type

1) has three stages and nine encounters, in four different

places. Then the listing starts for the next mission with

h=l, i=2, and continues until all missions for all types of

submarines have been described in this manner. Note that

this listing has already taken into account the classification

25

of the places k by £(k). This is done in such a way that

for each encounter on this mission listing the associated

C, •

ps, s is positive. In other words, a place k, and hence

an encounter, will appear on the mission listing only if it

seems possible that the opponent will allocate forces to that

place and the corresponding effectiveness against the maxi-

mized s submarine operating in that place is positive.

There are various possible ways of storing the

information contained in the mission description in the

machine. One way which is very economical in terms of

storage space requirements is to read the list encounter by

encounter, i.e., card by card, starting with an N=l. Then

A(N) = E'C, .„,n and an indicator arrayh3&(k) ;

P(N) = I-J-K-M- (h-1) + J-K-M-(i-l) + K-M-(j-l)

+ M« (k-1) + m + 1

contain the complete information. I, J, K, M denote the

maximum values of the indices i, j, k, m. The reason for

using (m+1) is that the correct m appears after the mth

stage is completed. The disadvantage of this method is

that during computations the individual indices must be re-

computed from P(N):

[•] means "the largestinteger in •"

h = 1 +P(N)

I • J • K • M

a = p(n; - I-J-K:-M(h-l)

i = 1 +A

J • K • M

AA = A J-K-M- (i

26

1 + _MK-M

AAA = AA - K«M(j -1)

k = 1 +AAAM

m = AAA - M(k-l)

In the sample program contained in this paper, the indices

in their original form are stored in vector arrays and are

directly accessible throughout the computations.

2. The Contraction with Respect to Vu .

^_him

This contraction takes into account the possibility

that the stage m of a mission may have been noted, but that

the associated V may have been declared to be zero.nim J

C5j. C'dlatS r.nythlii; IiVO i ViU^'him " '

the corresponding elements in the index arrays of the mission

are eliminated. This is done prior to the execution of the

game and may therefore be referred to as the basic contrac-

tion .

3. Working in Sub spaces

It can be expected that most of the x, . and y.-.

will be on their boundaries at any stage, including the

approximate optimal solution. This suggests the idea of

eliminating computations involving variables which have

fixed values either temporarily or throughout the process.

To do this, one reduces the dimensions of the

spaces, thus saving machine time. One can redefine the

space so as to "freeze" the variables on the boundaries

leaving the remainder free to move.

27

Where in the program should redefinition of the

Y-space take place? The minimizer can hold the subspace

fixed and continue to move in that subspace until an ap-

parent minimum has been reached. At this point the al-

location corresponding to this minimum has to be checked for

optimality in the full space. This will require at least

one iteration through the minimization routine in the full

space. After a minimum valid in the full space has been

obtained, the Y-space is reduced to a new subspace. An

alternative approach is to redefine the space after each

change in the Y-allocation , thus maintaining a continously

changing subspace. The authors feel that the latter will

enable the minimizing variable to stay in the subspace

icnger finding directions oi decicdse , before the ;iccd

arises to employ the full space. This idea. was implemented

in the program.

The contractions for x (statement 1871) and y (125)

eliminate computations involving elements that are on the

lower boundaries. It must be noted that the contraction

itself and the complications caused by its use take machine

time. Worthwhile time savings in computation will not be

realized until this technique is applied to relatively large

scale problems.

INDX(N), N = 1,...,NNX, contains the indices of

the xhi

> 0, INDY(N) , N = 1,...,NNY, contains the indices of

the y.-, > 0. NNX and NNY are the dimensions of the subspaces3 K

for X and Y. These index arrays are used to control which

28

variables are to be involved in the computations. The way

this is done in practice can be seen in the program list-

ing.

One remark concerning the DIRECTION FINDING ALGO-

RITHMS, (400) and (1400), may be in order: these algorithms

have to be applied row by row to the matrices | x.. | and

|y.,|

j. In subspaces, the number of elements belonging to3 K

a particular row varies. The outer do-loop runs over the

number of rows. To find the numbers of elements per row

(these numbers serve as the termination values of the inner

do-loops) a test on the indices stored in INDX and INDY is

conducted (402 f f ) , (1402 f f) . The test value, NTEST, is

the index of the last element in the rows of |x, . I andii hi ''

• y • - i i - i esDect ivelv . Wr> & r< the t-~s' rv<ass&s 313 c 1 ctjigti t sI .

j K . , . I > r - >

in the row at hand have been collected and the algorithm

begins. Before the next row is picked, NSTART is incre-

mented by the number of elements found in the previous row

so that the search through INDX or INDY always starts in the

correct position.

4 . Machine Accuracy Problems

The program calls for frequent testing of floating

point numbers. Throughout the development of the program

these tests have been sources of trouble. Testing for

equality must be strictly avoided even after - as has been

done here in various places - variables close to a fixed

quantity have been reset to that quantity (e.g., (1050 ff)

or (1060 f f ) ) . Although the program specified double-pre-

cision, it took extensive experimentation to maintain

29

feasibility, i.e., satisfy the side conditions

fxhi £ y jk - l

to at least the 13 tn decimal place.

Of critical importance is the accuracy in the

computation of the direction matrices y (GAX) and g (GAY)

.

In the neighborhood of a maximum or minimum, the partials

F and F are close to zero, as are the Lagrange multi-x y ' & &

pliers (XMU and YMU ) , and the differences F -XMU,F -YMU

(SD) . In order to determine the value of the derivative

game, the sum of the squares of these differences has to be

formed, an operation that may very likely lead to erroneous

results which, in turn, carry over into the computation of

y and g.

The countermeasure taken is to premultiply the SD

when they are small by a large number, (505 f f) , (1510 f f) .

5 . Testing the Program

As the calling of subroutines is exceedingly time

consuming, the present program does not use them. This made

debugging tedious. The "standard" routine, i.e., the pro-

gram employing the derivative games in the original form,

was tested running a small scale example where |x| was

(2 X 2) and |y| was (2 X 4), making it possible to hand-

check the computations.

The "auxiliary game" routine using the B-R process

was debugged using the following objective function:

30

. ^ ~ ~ „ ,1

-*- >> s w a

F(x,y) = 2x1

exp[ -2y1-y

2] + 2x

2exp[ -y

1"2y

2]

+ x3

exP [-y1-y

2-y

3 ]

Subject to: L, x. = Zj y, = 1

i k

xi> ^k *

°

with initial allocations x = (0,0,1), y=(0,l,0). For a

discussion of this example in light of the algorithm see

[4]. The reason that this can only be solved through the

B-R routine lies in the fact that, against the initial x,

any y represents an exact minimum, however, the value of

the derivative game for x is positive, yielded by y° =

(0, /2/2, -/2/2).

Pi ri 3 1 1 v pti <=>y -jiiit->1o i.i i f n ^ ^ Q £" COUnt" T2 WHS

up where |x| was (4x4), |y| was (5x10). While this does

not yet represent a problem in high-dimensional space, the

authors are confident that this example provided a suffi-

ciently severe test to demonstrate the validity of the pro-

gram as written.

6 . General Comments

The following remarks are intended to facilitate

the use and modification of the program.

a. Initial Allocation

The initial allocations are generated as corner

points in the stationary part (99). It can be expected that,

in the optimal solution, most of the variables will be on

the boundaries. An initial point on the boundaries insures

that the number of "absurd" allocations is minimal.

31

If one were to use an interior point as a

starting solution, it possibly would involve large numbers

of absurd allocations (e.g., a submarine in an aircraft

barrier patrol) . The machine would spend an enormous

amount of time reducing such allocations to zero.

b. Distance Policy

The initial and re-set values for the dis-

tances are determined from the dimensions of the spaces;

the user may employ his own rules. A compromise between

re-set values too large causing "overshooting" and values

to small causing "creeping" should be considered. In gen-

eral, "creeping" is much costlier in terms of machine time.

The policy for halving distances and its rationale is out-

lined in [ 4 ] .

c. Upper and Lower Bounds

For simplicity, and 1 have been used in the

program. Specifying individual bounds a, •, 3i • and a.-,

,

b., does not introduce additional problems but increases the

storage space required considerably (e.g., for y, two addi-

tional arrays of the same dimension as y)

.

,

d. Modifying the Objective Function

The algorithm as stated is valid for concave-

convex functions under quite general conditions. Though the

present program has been designed to handle a particular lin

ear-exponential function, it is quite flexible. The objec-

tive function mentioned in subsection 5 may serve to illus-

trate this point:

32

F(x,y) = 2x1exp[-2y

1-y

2]

+ 2x^ exp [-y1"2y

2]

+ x3

exp [-yr y 2-y

3]

is produced by a very simple adjustment of the mission

description

:

h=l i=l k=l j=l E=2.0k=2 j=l E=1.0k=3 j=l E=0.0 END OF MISSION 1

m=l

h=l i=2 k=l j=l E=1.0k=2 j=l E=2.0k=3 j=l E=0.0 END OF MISSION 2

m=l

h=l i=3 k=l j=l E=1.0k=2 j=2 E=1.0k=3 j=3 E=1.0 END OF MISSION 3

m=l

/•* r\

Vln = 2.0

v131

- 1.0.

The associated C, . n ,-. ^ are : C nin =Ci 1o = C 11 _=1.0hj£(k) 111 112 113

e. Assigning Values V, .& & him

The values (of completing stage m of mission i

for submarine of type h) are relative measures. When as-

signing V, . , the user should consider that a submarine mayfo ^ him' '

have spent part of its weapons (resources) during stage m-1

This may reduce its operational capabilities for stage m,

and the value for stage m should reflect this fact.

f. Choosing p (RHO)

The parameter p, mentioned in Chapter II,

section B 2.b, has to be chosen by the user. It specifies

33

the accuracy to which Max <}>(x) is to approximate the value

of the game F(x,y) , V. p affects the y-minimization by

controlling the accuracy, e(d), to which the derivative game

D Max F(x,y) = -VF -g has to approximate its mathematicalg x Y

value. The y-minimization is the most time consuming por-

tion of the process.

where 6 is the diameter of the X-space. Considering the

"worst" case, when d = d = p/36-L, (L is the maximum os-

cillation of F ) , it becomes apparent that

2

e ( do^=

36 2 -6-L

is of order of magnitude p2 *10

Recall that the conditions established for

the valididty of the algorithm are intended to guarantee

that, at the approximate optimal solution,|§ (x) -V| <p

.

Test runs of the program seem to indicate that the accuracy

actually obtained is much higher.

Although generally valid conclusions cannot be

derived from this observation the user must be aware of this

feature because he will pay heavily in terms of machine time

when p is unreasonably small.

34

IV. SUMMARY

The devlopment of the mathematical theory underlying

the derivative game and the concave -convex game algorithm

has only been sketched out in this paper. Here, that al-

gorithm has been employed in the formation of a potentially

useful example. With programming techniques designed to

provide economical running in high dimensions, large-scale

problems should be amenable to the application of the con-

cave-convex game algorithm.

35

V. SUGGESTIONS FOR FURTHER STUDY

A. MATHEMATICS

In Section III.C.6.e., the question of the proper choice

of p was addressed. Recall that

Max <f>(x) " V|< p.

x

Even when the user has based his choice of a value for p

on extensive experimentation with a program, he still will

not know exactly how close Max <{>(x) is to the value of thex

game. The desirable state of affairs would be

Max <Kx) _ V| = px

This would require the formulation of necessary conditions

on the functions a 3 and e

.

5

B. PROGRAMMING

The fact that, for the class of games at issue, MaxMinFx y

= MinMaxF can be exploited to arrive at the neighborhood ofy x

the optimal solution faster than the present program will:

Start the problem as MinMaxF. The course of action then is

reversed with considerable advantages. The maximize r sees

the present y-allocation ; the maximization is trivial, as-

signing as much weight as possible to the x, . with the

largest coefficients (the F,(x,y )), which results in a

corner solution for x. Then the B-R technique is employed

directly to F(x,y) which will bring x off the boundaries

36

again, and, after a pre-set number of iterations, will pro-

duce an allocation not far from the solution. At this stage

the problem is reinterpreted as MaxMinF and solved in the

manner of the present program.

Another feasible refinement particularly useful in the

application of the algorithm to objective functions linear

in both x and y, i.e., F = X x.a..y., is the idea of13 J J

"doubling", outlined in [4]. The problem is treated as

MaxMin and MinMax at the same time. Here the value of the

game is approached from below and above simultaneously which

provides a stopping rule when the difference MaxF(x,y) -cj) (x)x

arrives at a pre-specif ied value.

37

APPENDIX FLOWCHART OF THE PROGRAMMED EXAMPLE

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50

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BIBLIOGRAPHY

1. Danskin, J. M. , "Fictitious Play for Continuous Games",Naval Research Logistics Quarterly, 1, 1955

, pp. 313-3TCT

2. Danskin, J. M. , "The Theory of Max-Min, With Applica-tions", Journal of the Society of Industrial and Ap-plied Mathematics , v. 14, I960, pp. 641-664.

3. Danskin, J. M. , The Theory of Max-Min, Springer-Verlag, New YorF] inc

., 19 6 7.

4. Danskin, J. M. , "The Derivative Game in Max-Min andthe Solution of Concave-Convex Games", to appearJournal of the Society of Industrial and AppliedMathematics

.

5. Dresher', M., Games of Strategy, Theory and Applications,Prentice-Hall^ Englewood Cliifs, New Jersey, 1961.

6. Isaacs, R. , Differential Games, John Wiley and Sons,Inc., New York, I96S.

7. Kakutani, 5., "A Generalization of Brou"pr r

s FixedPoint Theorem", Duke Mathematical Journal, v. 8,No. 3, 1941, pp.~T57-4b9.

8. Karlin, S., Mathematical Methods and Theory in Games,Programming and Economics, Vols . I and 1 1 , Addison-Wesley , Reading, Massachus e 1 1 s , 19 59.

9. Koopman , B. 0., Search and Screening, OperationsEvaluation Group Report No. 56 , Washington , D. C.,1946.

10. Kuhn, H. W. , and Tucker, A. IV., (eds.), Contributionsto the Theory of Games , Vol. I, Annals o± MathematicsStudy No. 24 , Princeton University Press, Princeton,New Jersey , 1950

.

11. Morse, P. M. , and Kimball, G. W., Methods of OperationsResearch , John Wiley and Sons, Inc

., New York , 1951

.

12. Robinson, J'., "An Iterative Method of Solving a Game",Annals of Mathematics , v. 54, 1951, pp. 296-301.

13. Schroeder, R. G., Mathematical Models of AntisubmarineTactics , Ph.D. Thesis, Northwestern University, Evanston

,

I llinois , 1966

.

74

14. Schroeder, R. G., Search and Evasion Games , Naval Post-graduate School Technical Report/Research Paper No. 68,Naval Postgraduate School, Monterey, California, 1966.

15. Suppes , P., and Atkinson, R. C., Markov Learning Modelsfor Multiperson Interactions , Stanford University Press,Stanford, California, 1960.

16. Taylor, J. G., Application of Differential Games toProblems of Military Conflict: Tactical AllocationProblems - Part I , Naval Postgraduate School, Monterey,California, 1970.

17. Taylor, J. G., Application of Differential Games toProblems of Naval Warfare: Surveillance - EvasionPart I , Naval Postgraduate School, Monterey, California

,

1970.

18. von Neumann, J., "Zur Theorie der Gesellschaftsspiele"

,

Mathematische Annalen , Vol. 100, 1928, pp. 295-320.

19. Valentine, F., Convex Sets , McGraw-Hill, New York,1964.

20. Weyl, H. , "Elementary Proof of a Minimax Theorem Dueto von Neumann" , Contributions to the Theory ofG.-imijb , Vol. T

;;\nr:ai" or Mathematics Study No . 24,

Princeton University Press, Princeton, New Jersey,1950, pp. 19-25.

21. Williams, J. D., The Compleat Strategyst , McGraw-Hill,New York, 19 54.

22. Zmuida, P. T., An Algorithm for Optimization of CertainAllocation Models

,M.S. Thesis, Naval Postgraduate

School, Monterey, California, 1971.

75

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Documentation CenterCameron StationAlexandria, Virginia 22314

2. Library, Code 0212Naval Postgraduate SchoolMonterey, California 93940

3. Chief of Naval PersonnelPers-llbDepartment of the NavyWashington, D. C. 20370

4. Department of Operations Researchand Administrative Sciences

Naval Postgraduate SchoolMonterey, California 93940

5. Professor W. P. Cunningham, Code 55Cmm:"isv"r;,,-;.? of OueratiOiiS Research ar,.i

i

Administrative SciencesNaval Postgraduate SchoolMonterey, California 93940

6. Professor J. M. Danskin, Code 55DkDepartment of Operations Research andAdministrative Sciences

Naval Postgraduate SchoolMonterey, California 93940

7. LT Lee Fredric Gunn , USN10002 Enford CourtFairfax, Virginia 22030

8. Kapitanleutant Peter Tocha, German Navyc/o Alo Terhoeven415 KrefeldMarienstr. 43Germany

76

UNCLASSIFIEDSecurity Classification

DOCUMENT CONTROL DATA -R&D(Security clas si fie ation of title, body of ahstract and indexing annotation must be entered when the overall report is classified)

1 ORIGINATING ACTIVITY (Corporate author)

Naval Postgraduate SchoolMonterey, California 93940

2a. REPORT SECURITY CLASSIFICATION

UNCLASSIFIED?b. GROUP

3 REPOR T TITLE

AN ALGORITHM FOR THE SOLUTION OF CONCAVE -CONVEX GAMES

4 DESCRIPTIVE NOTES (Type of report and, inc Ius i vr dates)

Master's Thesis; September 19715 AUTHORISI ffifst name, middle initial, last name)

Lee F. GunnPeter Tocha

6. REPOR T D A TE

September 197170. TOTAL NO. OF PAGES

78

7b. NO. OF REFS

226a. CONTRACT OR GRANT NO.

b. PROJEC T NO

9a. ORIGINATOR'S REPORT NUMBER(S)

9b. OTHER REPORT NO(S) (Any other numbers that may be assignedthis report)

10 DISTRIBUTION STATEMENT

Approved for public release; distribution unlimited.

11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Naval Postgraduate SchoolMonterey, California 93940

13. ABSTRACT

A master's thesis which discusses the solution of concave-convex games. An algorithm is developed, a computerprogram written and applied to an anti-submarine warfareforce allocation problem as an illustration. Techniquesfor handling concave- convex problems in high dimensionsare included.

DD I NOV «!l*» / OS/N 0101 -807-681 1

(PAGE 1

)

UNCLASSIFIED77 Security Classification

A-31408

UNCLASSIFIEDSecurity classification

KEY WORDS

Concave-Convex Games

Games

ASW Forces

ASW Force Allocations

Game Theory

Resource Allocations

Allocations

Allocation Problems

Concave-Convex Functions

Derivative Games

Brown-Robinson Process

MaxMin

Directional Derivative

Algorithms

ROLE

DD .'•"."..1473 < BA«

LINK C

ROLE

B/N 0101-807-61JNCLASSIFIED

78 Security Classification A- U 409

TL-e <*7 hot l^tJU!

2 AUG72 DiJJDERY20906

ThesisG8647 Gunn

131297

c.l Anthe !

algorithm forsolution of con-

cave--convex games.

2 AU072D 1 HUSKY209 06

ThG8c:

Thesis

G8647c.l

Gunn 131297An algorithm for

the solution of con-cave-convex games.

3 2768 002 13596 4

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